A Little Deep

Here we will dig into a few topics a little deeper than normal. Some information here might not be relevant to your ops but...Remember - You never know too much.

​The Basis for 60-to-1

The concept of 60-to-1 says the number 60 has magical powers in the pilot world and there is a case to be made for that.An engineer will tell you that AFIIC stretches the 60-to-1 concept a bridge too far; the magical 60-to-1 idea doesn't actually explain anything. The rules of thumb do not result from simply multiplying or dividing with multiples of the number 60. The real explanation behind each technique has more to do with trigonometry in most cases.

Circumference of the earthContrary to common mythology, the idea that the earth is round predates Columbus. An early Greek scholar, Eratosthenes (276 BC - 195 BC) knew that the sun shone to the bottom of a well in the town of Syene (present day Aswan) on the summer solstice, and was therefore directly overhead. And yet it was not directly overhead in Alexandria, just 925 kilometers directly to the north.Eratostenes realized the sun's rays reach the earth in virtually parallel lines because of its distance. He measured the angle from vertical of the sun's rays in Alexandria when they were vertical in Syene to be 1/50th of a circle. He rationalized that the circumference of the earth would be 50 times the distance between the cities. Remarkably, he was accurate to within 0.4%.Of course we know the earth is not a perfect sphere, it is wider at the equator than it is north-to-south. We will use 21,654 nm (24,902 statute miles) for the purposes of the computations to come.

360° in a circleNobody really knows why there are 360° in a circle, other that a few hypothesis that all sound about right. Ancient astronomers, perhaps, realized each year seems to repeat itself after about 360 days and that the earth, therefore, moved 1/360th of its path around the sun every day.Latitude​Greek astronomer Claudius Ptolemy wrote about grids that spanned the earth in a treatise he called "Geography." He cataloged places he knew of in relation to the equator (north and south) and the Fortunate Islands (east and west). The system of using degrees north and south for latitude remain with us to this day. The system east and west still exist, of course, though based on a different location. (The Fortunate Islands are now the Canary Islands and Madeira.)

Simple division

The fundamental 60-to-1 theory comes from the following:

The earth is round and has a circumference of 21,654 nautical miles.

Because the earth is round, it can be divided by 360 to produce degrees of latitude.

21,654 / 360 = 60.15 nautical miles per degree of latitude.

A pilot can say 1 degree of latitude equals 60 nautical miles.

Of course this is off by 0.25%. Close enough!

60-to-1, The School Solution

Any international pilot worth the title knows that 1 degree of latitude equals 60 nautical miles, as proven above. From there we come up with the 60-to-1 theory itself.

Trigonometry

The Greek word for triangle is "trigonon" and from that we get the study of triangles, trigonometry. It is a subject that makes many pilots wince. In fact, you could argue many pilots became pilots because their high school math classes convinced them they should do something fun for a living, rather than spend their days writing formulas and drawing three angles surrounded by three connected lines. That is unfortunate; much of aviation is based on trigonometry.When you constrain one of the angles in a triangle to being precisely 90°, a right angle, you can learn a lot about the other parts of the triangle with relative ease. If you draw a circle around the triangle with one point at the center and another at the circumference, the tangent of the circle intersecting the triangle has a few interesting properties.

​We draw triangles and label the sides with lower case letters. The angle that is opposite that side is labeled by the same letter, in upper case. Just to make things a bit more confusing, we often label the angles using letters of the Greek alphabet, the most common being the letter theta, θ.The tangent of a triangle is found by dividing the side opposite that angle by the side adjacent to that angle. In a classic mathematical sense, the answer would be presented in radians (of which there are 2π in a circle) but for most uses degrees are preferred.tanA=side opposite Aside adjacent A=abtanA=side opposite Aside adjacent A=abFor example, let's say we have a triangle ABC where a = 1 and b = 2. Using a scientific calculator we see that the tangent of A = 0.1. Now that hardly seems useful, does it?We can make this function more useful if we could solve for A. This is known as an "inverse function" and the solution for A, in this case, is called the "arc tangent." It can be written as arctangent(A), arctan(A), atan(A), or more properly, tan-1(A).A=arctan(ab)A=arctan(ab)Our example becomes A = arctan( a / b ) = arctan ( 1 / 2 ) = 27°.So that is all you really need to know. Just keep in mind this formula converts two sides of a right triangle into an angle. The rest is easy.

Circle trigonometry, from Stephen Johnson

Examples

​CrosswindsJust as the sine of 30° = 0.5, the cosine of 60° = 0.5. You can also figure that the sine of 60 or the cosine of 30° is √3 / 2 = 0.87, almost 90 percent. The cosine or sine of 45° is 1 / √2 = 0.71, almost three-fourths. That leads to a few handy rules of thumb when it comes to crosswinds:

A thirty degree crosswind is equal to one-half the full wind factor.

A forty-five degree crosswind is equal to three-fourths the full wind factor.

A sixty degree crosswind is equal to ninety percent of the full wind factor.

​Crosswind chart

Course DeviationThe tangent of an angle provides the relationship of a triangle's two legs adjacent to the right angle. Multiplied by the distance left to travel, the tangent of an angle will provide course deviation.

Course azimuth deviation

Circling OffsetThe sine of 30° = 0.5, which offers us easy math for many situations. When approaching a runway to circle to the opposite side, for example, one often offsets 30 degrees to establish a downwind. The distance offset is equal to half the distance covered in the offset leg.

30° Circling offset

Vertical Navigation on Approach

We often use 300 feet per nautical mile as a wag for how high we should be on final approach. It is pretty close to a three degree glide path, convenient eh?It is more than convenience, it is trignonometry. The real number, as it turns out, is 318 feet per nautical mile:Height=6076(tan3°)=318feet

AGL vs. DME to go

​Radians

​Converting Radians to Degrees and Degrees to Radians

Just like a distance D can be measured in feet D (ft) or nautical miles D (nm), an angle θ can be measured in degrees θ (deg) or radians θ (rad). Like degrees, a radian is defined in relation to the properties of a circle. In particular, an angle θ (rad) is defined as the ratio of the length of a section of the circles’ circumference (arc length S) to the radius R as shown in the figure.Since we are going to be talking about angles with the Greek symbol theta, θ, with two different units, we will apply a subscript to differentiate between angles measure in radians, θRadians, and degrees, θDegrees.θRadians=SRIf the length of S happens to equal R, thenθRadians = S/R = R/R = 1 radian.

If S is twice the length of R, thenθRadians = S/R = 2R/R = 2 radians.

Now if we let S increase to the length of the circle’s circumference thenS = 2πRand θRadians = S/R = 2πR/R = 2π for a full circle.

So we conclude a complete circle represents 2π radians. But we also know a circle represents 360 degrees. Thus360 deg = 2π rads

We now can convert back and forth between degrees and radians:θRadians=θDegrees(π/180)and:θDegrees=θRadians(180/π)

Approximating Trigonometric Functions for Small Angles​For small angles it is often useful to approximate that the sine or tangent of the angle θRadians is equal to the angle itself, in radians. In other words, sin(θRadians) = tan(θRadians) = θRadians.Consider the diagram of the circle and right triangle, which is one way to visualize the small angle approximations Sin(θ) = Tan(θ) = θ in radians. CB and AB represent line segments.Tan(θ) = AB/R but for small θ we can say AB ≈ S, so we have:Tan(θ) = S/R and by definition S/R is θ in radians.Likewise we have:Sin(θ) = AB/CB but for small angles AB ≈ S and CB ≈ R, so we have:Sin(θ) = S/R and again S/R is θ in radians. Another conclusion is that for small angles:Tan(θ) = Sin(θ) although the percent error is a bit different.